Photonic Band Structure Calculation of System Possessing Kerr Nonlinearity

Abstract

The Finite Difference Time Domain (FDTD) technique dates back to 1966 when it was first developed by Yee [1]. Since then it has been widely used to calculate the radar cross section of objects as well as normal modes of wave guides. Chan et al. [2] recently applied this technique to calculate the band structure of photonic crystal with excellent results. The motivation for their use of this technique is the fact this technique scales linearly with system size. Systems having random defects, which destroy any periodicity, are often studied using the “super cell” method where the system is assumed to be periodic with a very large period. In such a case, it is important the technique for band structure calculation scales favorably with system size. The plane wave expansion technique, for example, is impractical since it scales as N³ where N is the system size. In this paper I show how the FDTD technique can be extended to calculate the band structure of nonlinear photonic crystals, in particular, those which possess Kerr nonlinearity.